At time `t=0`, particle `A` is at `(1m,2m)` and `B` is at `(5m,5m)`. Velocity of `B` is `(2hati+4hatj)m//s` Velocity of particle `A` is `sqrt(2)v)` at `45^(@)` with `x`-axis. A collides with `B` .
Time when `A` will collide with `B` is …….second.

Options

A. `0.5s`
B. `1.5s`
C. `4s`
D. `3s`

Answer

To solve the problem of when particle A will collide with particle B, we can follow these steps:

### Step 1: Determine the initial positions of particles A and B.
- Particle A is at the position \( (1 \, \text{m}, 2 \, \text{m}) \).
- Particle B is at the position \( (5 \, \text{m}, 5 \, \text{m}) \).

### Step 2: Determine the velocities of particles A and B.
- The velocity of particle B is given as \( \vec{V_B} = 2 \hat{i} + 4 \hat{j} \, \text{m/s} \).
- The velocity of particle A is given as \( \sqrt{2}v \) at an angle of \( 45^\circ \) with the x-axis. We can break this down into components:
- \( V_{Ax} = \sqrt{2}v \cos(45^\circ) = \sqrt{2}v \cdot \frac{1}{\sqrt{2}} = v \)
- \( V_{Ay} = \sqrt{2}v \sin(45^\circ) = \sqrt{2}v \cdot \frac{1}{\sqrt{2}} = v \)
- Thus, the velocity of particle A can be expressed as \( \vec{V_A} = v \hat{i} + v \hat{j} \).

### Step 3: Calculate the relative velocity of B with respect to A.
- The relative velocity \( \vec{V_{BA}} \) is given by:
\[ \vec{V_{BA}} = \vec{V_B} - \vec{V_A} = (2 \hat{i} + 4 \hat{j}) - (v \hat{i} + v \hat{j}) = (2 - v) \hat{i} + (4 - v) \hat{j} \]

### Step 4: Determine the distance between A and B at \( t = 0 \).
- The distance \( d \) between A and B can be calculated using the distance formula:
\[ d = \sqrt{(5 - 1)^2 + (5 - 2)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \, \text{m} \]

### Step 5: Calculate the angle of approach.
- The angle \( \theta \) of approach can be calculated using the tangent function:
\[ \tan(\theta) = \frac{3}{4} \]
- This indicates the direction in which A is approaching B.

### Step 6: Set up the equation for collision.
- For A to collide with B, the relative velocity must be directed towards B. Thus, we need to ensure that the components of the relative velocity align with the direction of the distance vector from A to B.
- The distance vector from A to B is \( (4, 3) \).

### Step 7: Calculate the magnitude of the relative velocity.
- The magnitude of the relative velocity \( |\vec{V_{BA}}| \) is given by:
\[ |\vec{V_{BA}}| = \sqrt{(2 - v)^2 + (4 - v)^2} \]

### Step 8: Calculate the time until collision.
- The time \( t \) until collision can be calculated using:
\[ t = \frac{d}{|\vec{V_{BA}}|} = \frac{5}{|\vec{V_{BA}}|} \]

### Step 9: Solve for \( v \) such that \( |\vec{V_{BA}}| \) is maximized.
- To find the value of \( v \) that satisfies the conditions for collision, we can set up the equations based on the relative velocities and solve for \( v \).

### Step 10: Substitute \( v \) and calculate \( t \).
- After determining \( v \), substitute it back into the equation for \( |\vec{V_{BA}}| \) and calculate the time \( t \).

### Final Answer:
After performing the calculations, we find that the time when A will collide with B is \( 0.5 \, \text{seconds} \).

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